Let V be the set of all positive real numbers. Determine whether V is a vector space with the following operations. x + y = xy Addition cx = xc Scalar multiplication If it is, then verify each vector space axiom; if it is not, then state all vector space axioms that fail.
Let V be the set of all positive real numbers. Determine whether V is a vector space with the following operations. x + y = xy Addition cx = xc Scalar multiplication If it is, then verify each vector space axiom; if it is not, then state all vector space axioms that fail.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let V be the set of all positive real numbers. Determine whether V is a
| x + y | = | xy |
|
Addition |
| cx | = | xc | Scalar multiplication |
If it is, then verify each vector space axiom; if it is not, then state all vector space axioms that fail.
| STEP 1: | Check each of the 10 axioms. (1) u + v is in V. This axiom holds.This axiom fails.
(2) u + v = v + u This axiom holds.This axiom fails.
(3) u + (v + w) = (u + v) + w This axiom holds.This axiom fails.
(4) V has a zero vector 0 such that for every u in V, u + 0 = u. This axiom holds.This axiom fails.
(5) For every u in V, there is a vector in V denoted by −u such that u + (−u) = 0. This axiom holds.This axiom fails.
(6) cu is in V. This axiom holds.This axiom fails.
(7) c(u + v) = cu + cv This axiom holds.This axiom fails.
(8) (c + d)u = cu + du This axiom holds.This axiom fails.
(9) c(du) = (cd)u This axiom holds.This axiom fails.
(10) 1(u) = u This axiom holds.This axiom fails.
|
|---|---|
| STEP 2: | Use your results from Step 1 to decide whether V is a vector space. V is a vector space.V is not a vector space.
|
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